What Is The First Step In Solving The Equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$?A. Subtract $\frac{1}{3}$ From Each Side Of The Equation.B. Add 2 To Each Side Of The Equation.C. Combine Like Terms.D. Multiply Each Side Of The

Understanding the Basics of Linear Equations

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, and we will explore the first step in solving the equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form $ax + b = c$, where a, b, and c are constants, and x is the variable.

The Equation to be Solved

The equation we will be solving is $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$. This equation is a linear equation, and it can be solved using basic algebraic techniques.

The First Step in Solving the Equation

The first step in solving the equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$ is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, the like terms are $\frac{2}{3}x$ and $\frac{1}{3}x$.

Combining Like Terms

To combine like terms, we need to add or subtract the coefficients of the like terms. In this case, we need to add the coefficients of $\frac{2}{3}x$ and $\frac{1}{3}x$. The coefficient of $\frac{2}{3}x$ is $\frac{2}{3}$, and the coefficient of $\frac{1}{3}x$ is $\frac{1}{3}$. To add these coefficients, we need to find a common denominator, which is 3.

Finding a Common Denominator

To find a common denominator, we need to multiply the numerator and denominator of each fraction by the same number. In this case, we need to multiply the numerator and denominator of $\frac{2}{3}$ by 1, and the numerator and denominator of $\frac{1}{3}$ by 1.

Adding the Coefficients

Now that we have a common denominator, we can add the coefficients of $\frac{2}{3}x$ and $\frac{1}{3}x$. The coefficient of $\frac{2}{3}x$ is $\frac{2}{3}$, and the coefficient of $\frac{1}{3}x$ is $\frac{1}{3}$. To add these coefficients, we need to add the numerators and keep the denominator the same.

Simplifying the Equation

After combining the like terms, the equation becomes $\frac{3}{3}x + 2 = 5$. The coefficient of $x$ is now $\frac{3}{3}$, which simplifies to 1.

The Final Answer

The final answer is that the first step in solving the equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$ is to combine like terms. By combining like terms, we can simplify the equation and make it easier to solve.

Conclusion

In conclusion, solving linear equations requires a step-by-step approach. The first step in solving the equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$ is to combine like terms. By combining like terms, we can simplify the equation and make it easier to solve. This article has provided a step-by-step guide on how to solve linear equations, and it has highlighted the importance of combining like terms in the first step.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. One of the most common mistakes is to forget to combine like terms. Another common mistake is to add or subtract the wrong terms. To avoid these mistakes, it is essential to carefully read the equation and identify the like terms.

Tips and Tricks

When solving linear equations, there are several tips and tricks to keep in mind. One of the most important tips is to always combine like terms. Another important tip is to simplify the equation as much as possible. By simplifying the equation, we can make it easier to solve.

Real-World Applications

Linear equations have numerous real-world applications. In physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the behavior of markets. In engineering, linear equations are used to design and optimize systems.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. The first step in solving the equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$ is to combine like terms. By combining like terms, we can simplify the equation and make it easier to solve. This article has provided a step-by-step guide on how to solve linear equations, and it has highlighted the importance of combining like terms in the first step.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about solving linear equations. Whether you are a student or a teacher, this article will provide you with the information you need to master the skills of solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form $ax + b = c$, where a, b, and c are constants, and x is the variable.

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, the like terms are $\frac{2}{3}x$ and $\frac{1}{3}x$.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. In this case, you need to add the coefficients of $\frac{2}{3}x$ and $\frac{1}{3}x$. The coefficient of $\frac{2}{3}x$ is $\frac{2}{3}$, and the coefficient of $\frac{1}{3}x$ is $\frac{1}{3}$. To add these coefficients, you need to find a common denominator, which is 3.

Q: What is a common denominator?

A: A common denominator is a number that can be divided evenly by both of the denominators of the fractions. In this case, the common denominator is 3.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. Once you have a common denominator, you can add the numerators and keep the denominator the same.

Q: What is the final answer to the equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$?

A: The final answer to the equation $\frac{2}{3}x + \frac{1}{3}x + 2 = 5$ is $x = \frac{9}{3}$. This simplifies to $x = 3$.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include forgetting to combine like terms, adding or subtracting the wrong terms, and not simplifying the equation.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications. In physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the behavior of markets. In engineering, linear equations are used to design and optimize systems.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through example problems and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By understanding the basics of linear equations and practicing solving them, you can develop a deeper understanding of mathematics and improve your problem-solving skills. This article has provided a Q&A guide to help you master the skills of solving linear equations.

Final Thoughts

Solving linear equations is a fundamental concept in mathematics, and it has numerous real-world applications. By mastering the skills of solving linear equations, you can develop a deeper understanding of mathematics and improve your problem-solving skills.